On some accelerated optimization algorithms based on fixed point and linesearch techniques for convex minimization problems with applications
Tóm tắt
In this paper, we introduce and study a new accelerated algorithm based on forward–backward and SP-algorithm for solving a convex minimization problem of the sum of two convex and lower semicontinuous functions in a Hilbert space. Under some suitable control conditions, a weak convergence theorem of the proposed algorithm based on a fixed point is established. Moreover, we choose the stepsize of our algorithm which is independent on the Lipschitz constant of the gradient of the objective function by using a linesearch technique, and then a weak convergence result of the proposed algorithm is analyzed. As applications, we apply the proposed algorithm for solving the image restoration problems and compare its convergence behavior with other well-known algorithms in the literature. By our experiment, the algorithms have a higher efficiency than the others.
Tài liệu tham khảo
Aremu, K.O., Izuchukwu, C., Grace, O.N., Mewomo, O.T.: Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. J. Ind. Manag. Optim. 17(4), 2161–2180 (2021). https://doi.org/10.3934/jimo.2020063
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)
Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operator. Springer, New York (2007)
Bussaban, L., Suantai, S., Kaewkhao, A.: A parallel inertial S-iteration forward–backward algorithm for regression and classification problems. Carpath. J. Math. 36, 35–44 (2020)
Combettes, P.L., Pesquet, J.C.: A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1, 564–574 (2007)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)
Cruz, J.Y.B., Nghia, T.T.A.: On the convergence of the forward–backward splitting method with linesearches. Optim. Methods Softw. 31, 1209–1238 (2016)
Dunn, J.C.: Convexity, monotonicity, and gradient processes in Hilbert space. J. Math. Anal. Appl. 53, 145–158 (1976)
Hanjing, A., Suantai, S.: A fast image restoration algorithm based on a fixed point and optimization. Mathematics 8, 378 (2020). https://doi.org/10.3390/math8030378
Huang, Y., Dong, Y.: New properties of forward–backward splitting and a practical proximal-descent algorithm. Appl. Math. Comput. 237, 60–68 (2014)
Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)
Kankam, K., Pholasa, N., Cholamjiak, P.: On convergence and complexity of the modified forward–backward method involving new linesearches for convex minimization. Math. Methods Appl. Sci. 42, 1352–1362 (2019)
Lin, L.J., Takahashi, W.: A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity 16, 429–453 (2012)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)
Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Sér. A Math. 255, 2897–2899 (1962)
Nakajo, K., Shimoji, K., Takahashi, W.: On strong convergence by the hybrid method for families of mappings in Hilbert spaces. Nonlinear Anal., Theory Methods Appl. 71(1–2), 112–119 (2009)
Okeke, C.C., Izuchukwu, C.: A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces. Optim. Methods Softw. 34(6), 1168–1183 (2019)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Phuengrattana, W., Suantai, S.: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006–3014 (2011). https://doi.org/10.1016/j.cam.2010.12.022
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 17, 877–898 (1976)
Suantai, S., Kankam, K., Cholamjiak, P.: A novel forward–backward algorithm for solving convex minimization problem in Hilbert spaces. Mathematics 8, 42 (2020). https://doi.org/10.3390/math8010042
Tan, K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)
Thung, K., Raveendran, P.: A survey of image quality measures. In: Proceedings of the International Conference for Technical Postgraduates (TECHPOS), Kuala Lumpur, Malaysia, 14–15 December, pp. 1–4 (2009)